Mock AIME 3 2006-2007 Problems/Problem 2

Problem

If $abc - ab - bc - ca + a + b + c = 2008$ for positive integers $a$ , $b$ ,and $c$ ,then determine the minimum value of $a + b + c$ .

Solution

Subtract 1 from both sides to make $abc - ab - bc - ca + a + b + c - 1= 2007 \Longrightarrow (a-1)(b-1)(c-1) = 2007 = 3^2\cdot 223$ . From factorization, let $a = 4, b = 4, c = 224$ . Any other prime factorization would increase one of three to be much greater than 224, so this minimizes the sum of $a + b + c = 4 + 4 + 224 = \boxed{232}$ .

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Mock AIME 2 2006-2007

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Mock AIME 3 2006-2007 Problems/Problem 2

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